Now we show that the subsampled estimator provides a good approximation of the full sample estimator under suitable criterion. Our aim is to characterize the closeness between the full sample estimator \(\widehat{\Psi }\)  and the subsampled estimator \(\widetilde{\Psi }\) . In the literature of functional linear regression, strong consistency of the estimated regression function is not generally possible. The estimators are only guaranteed to converge in some weak norm such as the prediction error on a new sample [1]}, [2]}.
We choose to measure the difference \(\widetilde{\Psi }-\widehat{\Psi }\)  based on the full sample prediction error \(\Vert \mathcal {T}\widetilde{\Psi }-\mathcal {T}\widehat{\Psi }\Vert _N^2= \frac{1}{N} \sum _{n=1}^N \langle x_n, \widetilde{\Psi }-\widehat{\Psi }\rangle ^2\) . When only a fraction of the whole dataset is employed to estimate \(\widetilde{\Psi }\) , this metric partially gauges the quality of
\(\widetilde{\Psi }\)  by samples out of the subsampled (training) dataset.
Our theoretical target boils down to the analysis of the error \( \mathcal {T}\widetilde{\Psi }-\mathcal {T}\widehat{\Psi }\) . Lemma REF  below provides a decomposition of the subsampling error, the proof of which can be found in Appendix . Recall that \(x_n\)  can be represented as \(x_n = \sum _{r = 1}^{\infty }\hat{\sigma }_r\hat{\xi }_{nr}\hat{\theta }_r\)  due to the Karhunen-Loève expansion (REF ). The decomposition involves an operator \(\mathcal {T}^+:\mathbb {R}^N\rightarrow \mathcal {H}_X\) , which maps \(\mathbf {a}\in \mathbb {R}^N\)  to \(\mathcal {T}^+\mathbf {a} = \frac{1}{N}\sum _{n=1}^N a_n x_n^+ \in \mathcal {H}_X\) , and
\(x_n^+ = \sum _{r=1}^R \hat{\xi }_{nr} \hat{\theta }_r/\hat{\sigma }_r\)  is defined for each \(x_n\) .
